polynomial analogon for Fermat’s last theorem
For polynomials^{} with complex coefficients, there is an analogon of Fermat’s last theorem. It can be proven quite elementarily by using Mason’s theorem (1983), but the original proof (about in 1900) was based on methods of algebraic geometry^{}.
Theorem. For an integer $n$ greater than 2, there exist no non-constant coprime^{} polynomials $x(t)$, $y(t)$, $z(t)$ in the ring $\u2102[t]$ satisfying
${[x(t)]}^{n}+{[y(t)]}^{n}={[z(t)]}^{n}.$ | (1) |
Remark. For $n=2$, the equation (1) is in e.g. as
$${(2t)}^{2}+{(1-{t}^{2})}^{2}={(1+{t}^{2})}^{2}.$$ |
References
- 1 Serge Lang: “Die $abc$-Vermutung”. – Elemente der Mathematik 48 (1993).
Title | polynomial analogon for Fermat’s last theorem |
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Canonical name | PolynomialAnalogonForFermatsLastTheorem |
Date of creation | 2013-03-22 19:13:05 |
Last modified on | 2013-03-22 19:13:05 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 8 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 12E05 |
Classification | msc 11C08 |
Synonym | Fermat’s last theorem for polynomials |
Related topic | MasonsTheorem |
Related topic | WeierstrassSubstitutionFormulas |